## Advances in Operator Theory

- Adv. Oper. Theory
- Volume 4, Number 1 (2019), 251-264.

### Dominated orthogonally additive operators in lattice-normed spaces

Nariman Abasov and Marat Pliev

#### Abstract

In this paper, we introduce a new class of operators in lattice-normed spaces. We say that an orthogonally additive operator $T$ from a lattice-normed space $(V,E)$ to a lattice-normed space $(W,F)$ is dominated, if there exists a positive orthogonally additive operator $S$ from $E$ to $F$ such that $\vert Tx \vert \leq S \vert x \vert$ for any element $x$ of $(V,E)$. We show that under some mild conditions, a dominated orthogonally additive operator has an exact dominant and obtain formulas for calculating the exact dominant of a dominated orthogonally additive operator. In the last part of the paper we consider laterally-to-order continuous operators. We prove that a dominated orthogonally additive operator is laterally-to-order continuous if and only if the same is its exact dominant.

#### Article information

**Source**

Adv. Oper. Theory, Volume 4, Number 1 (2019), 251-264.

**Dates**

Received: 27 April 2018

Accepted: 2 August 2018

First available in Project Euclid: 20 September 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aot/1537408975

**Digital Object Identifier**

doi:10.15352/aot.1804-1354

**Mathematical Reviews number (MathSciNet)**

MR3867344

**Zentralblatt MATH identifier**

06946453

**Subjects**

Primary: 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) [See also 45Gxx, 45P05]

Secondary: 47H99: None of the above, but in this section

**Keywords**

Lattice-normed space vector lattice orthogonally additive operator dominated $\mathcal{P}$-operator exact dominant laterally-to-order continuous operator

#### Citation

Abasov, Nariman; Pliev, Marat. Dominated orthogonally additive operators in lattice-normed spaces. Adv. Oper. Theory 4 (2019), no. 1, 251--264. doi:10.15352/aot.1804-1354. https://projecteuclid.org/euclid.aot/1537408975